Is it true that for any finite cyclic group $G$, it holds that $\operatorname{exp}(G)=|G|$?
My first thought was yes, since if $G$ is cyclic, then we know it has an element of order |G|, and since any other $g\in G$, $\operatorname{order} (g)$ divides $|G|$ (from Lagrange's theorem), we have that $\operatorname{lcm}(g,|G|)=|G|$ for all $g\in G$.
But I wasn't sure, so I thought to ask, and I would also like to ask what other properties does the group exponent hold? (For some reason there isn't too much information about group exponent on the internet... not in Wikipedia, not in Proof Wiki, and certainly not in any pdf document I could find online...)
Many thanks in advanced.